The document discusses partitions of R3 (3-dimensional real space) into unit circles with and without the Axiom of Choice. It notes that without Choice, R3 can be partitioned into circles, but with Choice it can be partitioned into unit circles. The proof shows any partial partition into unit circles can be extended to a complete partition. However, sometimes there is not enough space to extend a partial partition. The document then discusses using Cohen forcing to construct a model where there is a partition of unit circles but no well-ordering of the real numbers. It presents lemmas on extending partial partitions and the transcendence degree when adding Cohen or other real numbers.